Talk:Geometric calculus

Possible references

• An Introduction to Geometric Algebra and Calculus by Alan BromborskySelfstudier (talk) 10:34, 2 February 2012 (UTC)
• A New Approach to Differential Geometry Using Clifford’s Geometric Algebra by John Snygg (2010) ISBN 978-0-8176-8282-8 — Quondum^☏✎ 08:55, 7 February 2012 (UTC)

Things to be sure to add

If I remember right (which is in all likelihood unlikely), the del operator is actually represented by an algebra element? And also invertible? Rschwieb (talk) 17:58, 6 February 2012 (UTC)

No, it is not an element of the algebra, but is an operator, just as the derivative is an operator in complex analysis. And yes, it is invertible. It will feature significantly, it won't be forgotten. — Quondum^☏✎ 18:09, 6 February 2012 (UTC)

Interior and exterior derivative

Hestenes is very careful to avoid calling $\nabla \wedge A$ the "exterior derivative" and $\nabla \cdot A$ the "interior derivative" or anything like that. He reserves these terms exclusively for those expressions that involve measures. That is, let ${\underline {L}}(B)$ be a linear map on directed measures (for example, ${\underline {L}}(dS)=dS\cdot F$ for some $n-1$-vector field), then the exterior differential is $\mathrm {d} {\underline {L}}={\underline {\dot {L}}}(dV\cdot {\dot {\nabla }})$, and similarly (trading a dot for a wedge) for the "interior differential". Hestenes likes coining new phrases or terminology when it suits him, though. Should we address the difference between, for lack of a better way to put it, the exterior derivative of a multivector field and the exterior derivative of a "form"? Muphrid15 (talk) 06:22, 21 May 2013 (UTC)

Some rather confusing terminology has been introduced into GA (and presumably GC), possibly by Hestenes. I'm not a fan of "outer product", "inner product" and the like because they conflict with the same terms used in closely related disciplines. I think that we should be careful to survey the literature for the terms used for a concept before adopting a particular author's choices. There is already a well-established term exterior derivative in an extremely closely related field, so I think that we should take care not to use it in a incompatible way, irrespective of Hestenes's use. — Quondum 00:28, 22 May 2013 (UTC)

Hestenes markets GC as an improvement over / replacement for the closely related disciplines. You're supposed to "unlearn" differential forms and all its jargon, then relearn GC and all its jargon. As this is the driving attitude, then, in my opinion, GC is allowed to trump terms like "exterior derivative". This isn't the theory of differential forms, so the definition of exterior derivative there isn't important here, so long as the reader isn't confused. I tried to clarify this a bit in the article ("Apart from a subtle difference in meaning for the exterior product with respect to differential forms versus the exterior product with respect to vectors..."). If this isn't enough, you can of course add your own clarifications. As a side note, I wrote the bulk of this article mostly following the reference linked near the top of this talk page, not Hestenes. Teply (talk) 18:51, 22 May 2013 (UTC)

Nice work!

Thanks to Teply for extending the interesting article. M∧Ŝc²ħεИ_τlk 23:41, 21 May 2013 (UTC)

Ideas for future expansion

Just throwing some ideas out here, for things that may be relevant for the future of this page.

• More differential geometry: the concept of vector manifolds, Hestenes' "method of mobiles" and/or Gull, Doran, and Lasenby's gauge theory gravity (which is relevant because it essentially provides a flat-space method of doing differential geometry)
• A table of geometric calculus identities, possibly with a comparison to similar vector calculus, tensor calculus, or differential forms identities. This may belong here, or may be better served by being a separate page.
• A section on how complex analysis is subsumed by geometric calculus--the theory of holomorphic functions and analyticity, meromorphic functions as real vector fields with delta function sources, GC renderings of common complex analysis theorems
• Multivector differentiation and applications: e.g. spinor differentiation for use in the Euler-Lagrange equation.

I'm not sure what Teply has in mind, so these ideas may or may not already overlap with what he's got planned. Muphrid15 (talk) 03:11, 22 May 2013 (UTC)

I'll address your bullet points in reverse order.

Multivector differentiation is obviously a major topic and should be included. Don't assume I have "plans," so just be bold and add it yourself. As always, if adding major content like that makes the article too long, then this page can be an overview and the subsections can become their own pages. The simple example of geometric calculus in the plane maps to complex analysis and could also be included here along with topics like the inverse of $\nabla$ and Green's functions. A table comparing different methods is usually helpful, though it is a kind of "meta" topic that deserves its own article.

Your first bullet point is a little trickier. The purpose of geometric calculus is, in some sense, to do better than differential geometry. I agree you can spend lots of time on the theory as it is in the GC language, but I would avoid getting too caught up in mapping everything back into the usual differential geometry language. You may want to reserve that for the "meta" article that has the comparison table. By the way, you mentioned gauge theory gravity. I started an article about GTG some time ago, but it's still kind of weak. If you know more about this topic, then you may want to focus your attention there. Teply (talk) 18:08, 22 May 2013 (UTC)

Apparent discrepancy on fundamental theorem?

Something that's been bugging me for a while that I thought must just be a subtlety I'd missed, but the article here (in agreement with Doran and Lasenby) gives the form for the fundamental theorem as depending on $\nabla \cdot dV$, but Hestenes and Sobczyk seem to give it as $dV\cdot \nabla$ (well, they use partial instead of del, but this is not a material difference). Is there actually a problem here that affects the accuracy of the article, or is it just an illusory difference? Muphrid15 (talk) 06:02, 4 June 2013 (UTC)

There is no discrepancy. The methods employed for determining of the relative orientations of the volume and the boundary are not the same, as you will see with a simple 2-dimensionnal Green's theorem. Stay with Doran Lasenby ; it is more general and thus safer. Chessfan (talk) 20:46, 2 September 2013 (UTC)

Proof of the independence on the choice of Frame?

Why is $e^{j}\partial _{j}$ independent of the choice of frame? It should be easy but still I can't manage to prove it.

If I take an other frame $(e')$, then I want to show that $e'^{j}\partial _{e'_{j}}=e^{j}\partial _{e_{j}}$. I write $e'_{j}=A_{j}^{k}e_{k}$, and then I use the linearity of the directional derivative with respect to the direction:

$e'^{j}\partial _{e'_{j}}=e'^{j}A_{j}^{k}\partial _{e_{k}}$

Then I just need to prove that $A_{j}^{k}e'^{j}=e^{k}$. To do that I need to prove that $(A_{j}^{k}e'^{j})\cdot e_{i}=\delta _{i}^{k}$, but I can't. What am I missing?--Grondilu (talk) 00:22, 7 September 2014 (UTC)

This relates to one of my pet peeves: the assumption of a holonomic basis without being explicit. Many people seem to be unaware that this is an arbitrary (albeit usually convenient) choice and not a necessary choice, and this has bled into WP. The answer to your question is straightforward: a holonomic basis is chosen so that this (or an equivalent) identity holds. (The linked article is actually misleading: the basis vectors are not equivalent to partial derivatives; they are chosen so that they transform in the same way.) —Quondum 03:06, 7 September 2014 (UTC)

I'm pretty sure $\partial _{i}$ in this article is the directional derivative operator in the direction $e_{i}$, so your concern does not apply. However you're right in the sense that the article currently suggests $\partial _{i}={\frac {\partial }{\partial x^{i}}}$, which is not defined in this article, and not true anyway (otherwise one might wonder what was the point of defining the directional derivative just before).

More precisely $\partial _{i}$ should defined here as:

$\partial _{i}:F\mapsto (x\mapsto e_{i}\cdot \nabla F(x))$

--Grondilu (talk) 03:36, 7 September 2014 (UTC)

Thanks for the edits – definitely headed in the right direction. Many people simply assume that $\partial _{i}$ is a shorthand for ${\frac {\partial }{\partial x^{i}}}$ rather than a directional derivative; so it is nice to see it defined as an operator. The article Holonomic basis will need cleaning up. The notation $b\cdot \nabla$ as an operator is confusing (and becomes notationally ambiguous once the geometric derivative has been defined), so I'd prefer to avoid its use. Does it make sense to use the more verbose notation $\partial _{e_{i}}$ or $\nabla _{e_{i}}$? Clarity is worth some verbosity. —Quondum 04:41, 7 September 2014 (UTC)

The ambiguity is raised provided it is proven that both meanings collide. If the proof can be given in the article I agree with replacing $b\cdot \nabla$ by $\partial _{b}$ in the article up until the point where it is proven that it really means the same as $b\cdot \nabla$. But without proof, I'm afraid we have to keep $b\cdot \nabla$ all along because it is such a notorious notation we just can't not mention it.--Grondilu (talk) 05:45, 7 September 2014 (UTC)

Once the geometric derivative has been defined, it can be introduced as composition of operators: the operator consisting of b dotting with a vector and that of the geometric derivative (and then one would in principle need the proof you mention). I agree with the approach of a different notation up to that point.

I'm getting into matters of taste now: the notations $\partial _{b}$ (subscripted vector) and $\partial _{i}$ (subscripted index). The latter is quite nice because it describes a genuine covariance as per the index, and it might be nice to use $\nabla _{b}$ for the interim notation for the directional derivative in a direction set by a vector other than a basis vector. This is just a thought (and I realize that I'm not being consistent); you're evidently more at home with this than I am. —Quondum 06:45, 7 September 2014 (UTC)

$\partial _{i}$ and $\partial _{b}$ are definitely not the same thing. $\partial _{i}$ is a short notation for $\partial _{e_{i}}$ which is the special case of $\partial _{b}$ where $b=e_{i}$, in other words it's the directional derivative along a direction indicated by one of the vectors of the basis, while $\partial _{b}$ is general case of the directional derivative in any direction $b$.--Grondilu (talk) 07:17, 7 September 2014 (UTC)

Understood. What I was suggesting was avoiding the notation $\partial _{b}$ (and hence also $\partial _{e_{i}}$) because it is so notationally similar to $\partial _{i}$ and hence potentially a source of confusion, and using $\nabla _{b}$ in its place. This allows us to formally define $\partial _{i}=\nabla _{e_{i}}$. Of course, we also have, in the end, $\nabla _{b}=b\cdot \nabla$, and $\partial _{i}=e_{i}\cdot \nabla$. The only notational partial collision we are left with then is the similarity between $\nabla$ and $\nabla _{b}$, but these are distinct enough (presence and absence of a subscript). There are two instances $\partial _{b}$ that would become $\nabla _{b}$, and two instances of $\partial _{e_{i}}$ that would become $\nabla _{e_{i}}$, if you agree. I'll modify as a trial; see how you feel about it. —Quondum 19:31, 7 September 2014 (UTC)

Well, $b$ is a vector and $i$ is a integer, so if there is confusion, it's related to the lack (in this article) of notation convention to distinguish vectors from scalars. Usually in the literature, vectors are noted either in boldface ($\mathbf {b}$) or with an upper arrow (${\vec {b}}$). Your suggestion seems to make sense, but I would very much prefer not to need it by picking a notation for vectors.--Grondilu (talk) 21:01, 7 September 2014 (UTC)

Geometric algebra (and hence geometric calculus) and much of WP math do not seem to use distinguishing notation for vectors, other than in physics articles meaning 3 dimensional Euclidean vectors. We are in a sense overworking symbols, in particular $\partial$. Would you feel more comfortable swapping the two (i.e. using $\partial _{b}$ and $\nabla _{i}$)? This would also avoid ambiguity with the oft-used interpretation of $\partial _{i}$ as ${\tfrac {\partial }{\partial x_{i}}}$. —Quondum 02:16, 8 September 2014 (UTC)

You're actually right about most literature not using different fonts for vectors. On the other hand, I've just checked for instance in Differential forms in Geometric Calculus from Hestenes, and he uses Greek letters for indexes. This can serve as a disambiguation between $\partial _{b}$ and $\partial _{\mu }$. More references should be looked at though before we decide something. Meanwhile I'm ok with using $\nabla$ whenever it's considered less ambiguous than $\partial$--Grondilu (talk) 08:33, 8 September 2014 (UTC)

Just verified: in his Oersted Medal Lecture (2002), Hestenes does use boldface font to differentiate vectors.--Grondilu (talk) 08:37, 8 September 2014 (UTC)

We should take care not to blindly follow Hestenes's early conventions, which were probably motivated by a need to explain grading of the algebra rather than to act as a reference (actually I'm averse to Hestenes's conventions: they're divergent and collide with others; it's a bit like deliberately inventing a new lexicon for a new field; he is also not the originator of the field). Other conventions occur with later authors, including the use of Greek letters for other purposes, e.g. for scalars, with indices being from any alphabet. Hestenes tended to provide special-case illustrations rather than general-case expositions, and his notation may have followed the convention used in some WP articles: the use of bold to mean specifically (3-d) Euclidean vectors, or to else distinguish a chosen spatial subspace of a Minkowski space. Use of distinguishing notation for different grades of the algebra is not appropriate in the general context, and so distinguishing vectors from other elements of the algebra is more of a problem than not. Since there is no established notation for the field, I would prefer a bias towards consistency with WP notation, but we should at least keep consistency between with Geometric algebra and related articles. Our readership is probably also not solely interested in GA/GC, which suggests value in maintaining more consistency across articles. I'll take a look at what references I have when I have them to hand. —Quondum 16:52, 8 September 2014 (UTC)

Claim on nonassociativity of exterior derivative

In Geometric_calculus § Differentiation §§ Interior_and_exterior_derivative, the following statement is made:

Unlike the exterior product, the exterior derivative is not even associative.

Does such a statement even make sense? First, the operator in question (denoted ∇∧) is an operator, not some element combined with the exterior product operator. Second, if one were to define an operator ∇∧∇∧, it would be defined as a composition of operators ∇∧ ∘ ∇∧, and it would inherently be "associative": (∇∧ ∘ ∇∧)A ≝ ∇∧ (∇∧ A). Or am I confused? —Quondum 04:55, 8 September 2014 (UTC)

I agree it does not make much sense, since the exterior derivative is not defined as a binary operation, so the concept of associativity does not apply. But maybe that's precisely what the author meant? In any case it is poorly phrased.--Grondilu (talk) 08:24, 8 September 2014 (UTC)

Thanks for the input. I've removed it, since it adds more confusion than anything. —Quondum 16:07, 8 September 2014 (UTC)

Derivative nomenclature

I'm changing the term "geometric derivative" to "vector derivative" everywhere it occurs in this article, for several reasons:

1) The term "geometric derivative" isn't used in any of the books I have on geometric calculus. Instead, the term "vector derivative" is used for what this article is calling the "geometric derivative". See, e.g., Clifford Algebra to Geometric Calculus by Hestenes and Sobczyk, p. 49, Geometric Algebra for Physicists by Doran and Lasenby, p. 168, and Vector and Geometric Calculus by Macdonald, p. 47.

2) The phrase "geometric derivative" sounds like it would mean the derivative in geometric calculus in general, but that's what the books call the "multivector derivative", of which the vector derivative is just a special case. This article only defines and discusses the derivative that the books call the "vector derivative", and doesn't discuss the broader multivector derivative at all.

3) The term "geometric derivative" is also used for the unrelated derivative used in multiplicative calculus, as explained in this Michael Penn video: https://www.youtube.com/watch?v=lQ_AdAFVsaM

Red Act (talk) 20:31, 8 December 2022 (UTC)

P.S.: I just encountered the first book I've ever seen that uses the phrase "geometric derivative" to mean "vector derivative", An Introduction to Geometric Algebra and Geometric Calculus by M.D. Taylor. However, on page 134 Taylor notes that "... what we have called the geometric derivative tends to be called the vector derivative...". This book was just published in 2021, so perhaps people sometimes using the phrase "geometric derivative" for this is a new phenomenon. Red Act (talk) 14:56, 3 August 2023 (UTC)

The term 'vector derivative' seems fine to me. If in 10 years the term 'geometric derivative' becomes common, we can revisit that decision. –jacobolus (t) 16:44, 3 August 2023 (UTC)